Optimal. Leaf size=96 \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]
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Rubi [A] time = 0.400461, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]
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Rubi in Sympy [A] time = 52.401, size = 83, normalized size = 0.86 \[ \frac{2 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{a d - b c} F\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}\middle | \frac{\left (e - 1\right ) \left (- a d + b c\right )}{a d}\right )}{b \sqrt{d} \sqrt{c + d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)
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Mathematica [A] time = 0.598351, size = 126, normalized size = 1.31 \[ -\frac{2 \sqrt{c+d x} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{(b c-a d) (e-1)}{a d}\right )}{d \sqrt{-\frac{a}{e-1}} \sqrt{\frac{b (e-1) x}{a}+e} \sqrt{\frac{b (c+d x)}{d (a+b x)}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]
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Maple [B] time = 0.171, size = 207, normalized size = 2.2 \[ 2\,{\frac{\sqrt{bx+a}\sqrt{dx+c} \left ( ade-bce+bc \right ) }{ \left ( d{x}^{2}b+adx+bcx+ac \right ) bd \left ( -1+e \right ) }\sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+e \right ) }{a}}}\sqrt{-{\frac{ \left ( dx+c \right ) b \left ( -1+e \right ) }{ade-bce+bc}}}{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \sqrt{c + d x} \sqrt{e + \frac{b e x}{a} - \frac{b x}{a}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="giac")
[Out]